Limit to Heisenberg in Temporal Physics

 Limit to Heisenberg 

consider

Δx⋅Δp≥ ℏ/2c

Modified Schrödinger Equation

In a potential-free region, the modified Schrödinger equation is:

i ℏ ∂ψ/∂t = -ℏ² / [2m (1 + α t²)] ∇²ψ

Deriving the Uncertainty Relation

Wavefunction Form:

Assume a plane wave solution for the wavefunction ψ:

ψ(x, t) = A e^(i (k x - ω t))

where k is the wave number and ω is the angular frequency.

Momentum and Energy Operators:

The momentum operator in quantum mechanics is:

p̂ = -i ℏ ∂/∂x

For a plane wave, the momentum is:

p = ℏ k

The kinetic energy operator is:

T̂ = -ℏ² / [2m (1 + α t²)] ∇²

For a plane wave, the kinetic energy is:

T = ℏ² k² / [2m (1 + α t²)]

Uncertainty in Position (Δx):

For a plane wave, the position uncertainty Δx is related to the wave packet's spatial extent. For simplicity, consider a Gaussian wave packet where:

Δx ≈ 1 / (√2 k)

Uncertainty in Momentum (Δp):

The uncertainty in momentum Δp can be derived from the relation:

Δp ≈ ℏ k

Uncertainty Product:

The product of uncertainties Δx ⋅ Δp is:

Δx ⋅ Δp ≈ [1 / (√2 k)] ⋅ (ℏ k) = ℏ / √2

Note: This derivation assumes a Gaussian wave packet for simplicity. For a more accurate calculation, integrate over the actual probability distribution of the wave packet.

Including the Modified Schrödinger Equation Scaling:

The modified uncertainty relation should take into account the scaling factor in the kinetic energy term:

Δx ⋅ Δp ≈ ℏ / [2] ⋅ 1 / (1 + α t²) = ℏ c / [2] ⋅ 1 / (1 + α t²)

Compare with the Modified Uncertainty Principle:

The modified uncertainty principle is:

Δx ⋅ Δp ≥ ℏ / [2c]

To ensure consistency:

ℏ c / [2] ⋅ 1 / (1 + α t²) ≥ ℏ / [2c]

Simplify:

c / (1 + α t²) ≥ c

Which is:

c ≥ c (1 + α t²)

To satisfy this condition, we need:

c / (1 + α t²) ≥ c

This implies:

1 / (1 + α t²) ≥ 1

Since α t² ≥ 0, this condition holds true if α t² is small or zero. In physical scenarios where α t² is small compared to 1, the inequality holds, ensuring that the modified Schrödinger equation is consistent with the modified uncertainty principle.

Comparison with Planck Time

Planck Time:

Value: 𝑡_𝑃≈5.39×10^−44

Relation: Planck time is a fundamental unit of time where quantum effects of gravity are expected to become significant.

Modified Uncertainty Bound:

Value: ℏ/2𝑐≈1.76×10^−43m⋅kg⋅m/s